src/HOL/Library/RBT.thy
author bulwahn
Tue Sep 13 09:28:03 2011 +0200 (2011-09-13)
changeset 44913 48240fb48980
parent 43124 fdb7e1d5f762
child 45694 4a8743618257
permissions -rw-r--r--
correcting theory name and dependencies
     1 (* Author: Florian Haftmann, TU Muenchen *)
     2 
     3 header {* Abstract type of Red-Black Trees *}
     4 
     5 (*<*)
     6 theory RBT
     7 imports Main RBT_Impl
     8 begin
     9 
    10 subsection {* Type definition *}
    11 
    12 typedef (open) ('a, 'b) rbt = "{t :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt. is_rbt t}"
    13   morphisms impl_of RBT
    14 proof -
    15   have "RBT_Impl.Empty \<in> ?rbt" by simp
    16   then show ?thesis ..
    17 qed
    18 
    19 lemma rbt_eq_iff:
    20   "t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2"
    21   by (simp add: impl_of_inject)
    22 
    23 lemma rbt_eqI:
    24   "impl_of t1 = impl_of t2 \<Longrightarrow> t1 = t2"
    25   by (simp add: rbt_eq_iff)
    26 
    27 lemma is_rbt_impl_of [simp, intro]:
    28   "is_rbt (impl_of t)"
    29   using impl_of [of t] by simp
    30 
    31 lemma RBT_impl_of [simp, code abstype]:
    32   "RBT (impl_of t) = t"
    33   by (simp add: impl_of_inverse)
    34 
    35 
    36 subsection {* Primitive operations *}
    37 
    38 definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" where
    39   [code]: "lookup t = RBT_Impl.lookup (impl_of t)"
    40 
    41 definition empty :: "('a\<Colon>linorder, 'b) rbt" where
    42   "empty = RBT RBT_Impl.Empty"
    43 
    44 lemma impl_of_empty [code abstract]:
    45   "impl_of empty = RBT_Impl.Empty"
    46   by (simp add: empty_def RBT_inverse)
    47 
    48 definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
    49   "insert k v t = RBT (RBT_Impl.insert k v (impl_of t))"
    50 
    51 lemma impl_of_insert [code abstract]:
    52   "impl_of (insert k v t) = RBT_Impl.insert k v (impl_of t)"
    53   by (simp add: insert_def RBT_inverse)
    54 
    55 definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
    56   "delete k t = RBT (RBT_Impl.delete k (impl_of t))"
    57 
    58 lemma impl_of_delete [code abstract]:
    59   "impl_of (delete k t) = RBT_Impl.delete k (impl_of t)"
    60   by (simp add: delete_def RBT_inverse)
    61 
    62 definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" where
    63   [code]: "entries t = RBT_Impl.entries (impl_of t)"
    64 
    65 definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" where
    66   [code]: "keys t = RBT_Impl.keys (impl_of t)"
    67 
    68 definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" where
    69   "bulkload xs = RBT (RBT_Impl.bulkload xs)"
    70 
    71 lemma impl_of_bulkload [code abstract]:
    72   "impl_of (bulkload xs) = RBT_Impl.bulkload xs"
    73   by (simp add: bulkload_def RBT_inverse)
    74 
    75 definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
    76   "map_entry k f t = RBT (RBT_Impl.map_entry k f (impl_of t))"
    77 
    78 lemma impl_of_map_entry [code abstract]:
    79   "impl_of (map_entry k f t) = RBT_Impl.map_entry k f (impl_of t)"
    80   by (simp add: map_entry_def RBT_inverse)
    81 
    82 definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
    83   "map f t = RBT (RBT_Impl.map f (impl_of t))"
    84 
    85 lemma impl_of_map [code abstract]:
    86   "impl_of (map f t) = RBT_Impl.map f (impl_of t)"
    87   by (simp add: map_def RBT_inverse)
    88 
    89 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
    90   [code]: "fold f t = RBT_Impl.fold f (impl_of t)"
    91 
    92 
    93 subsection {* Derived operations *}
    94 
    95 definition is_empty :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where
    96   [code]: "is_empty t = (case impl_of t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)"
    97 
    98 
    99 subsection {* Abstract lookup properties *}
   100 
   101 lemma lookup_RBT:
   102   "is_rbt t \<Longrightarrow> lookup (RBT t) = RBT_Impl.lookup t"
   103   by (simp add: lookup_def RBT_inverse)
   104 
   105 lemma lookup_impl_of:
   106   "RBT_Impl.lookup (impl_of t) = lookup t"
   107   by (simp add: lookup_def)
   108 
   109 lemma entries_impl_of:
   110   "RBT_Impl.entries (impl_of t) = entries t"
   111   by (simp add: entries_def)
   112 
   113 lemma keys_impl_of:
   114   "RBT_Impl.keys (impl_of t) = keys t"
   115   by (simp add: keys_def)
   116 
   117 lemma lookup_empty [simp]:
   118   "lookup empty = Map.empty"
   119   by (simp add: empty_def lookup_RBT fun_eq_iff)
   120 
   121 lemma lookup_insert [simp]:
   122   "lookup (insert k v t) = (lookup t)(k \<mapsto> v)"
   123   by (simp add: insert_def lookup_RBT lookup_insert lookup_impl_of)
   124 
   125 lemma lookup_delete [simp]:
   126   "lookup (delete k t) = (lookup t)(k := None)"
   127   by (simp add: delete_def lookup_RBT RBT_Impl.lookup_delete lookup_impl_of restrict_complement_singleton_eq)
   128 
   129 lemma map_of_entries [simp]:
   130   "map_of (entries t) = lookup t"
   131   by (simp add: entries_def map_of_entries lookup_impl_of)
   132 
   133 lemma entries_lookup:
   134   "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
   135   by (simp add: entries_def lookup_def entries_lookup)
   136 
   137 lemma lookup_bulkload [simp]:
   138   "lookup (bulkload xs) = map_of xs"
   139   by (simp add: bulkload_def lookup_RBT RBT_Impl.lookup_bulkload)
   140 
   141 lemma lookup_map_entry [simp]:
   142   "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
   143   by (simp add: map_entry_def lookup_RBT RBT_Impl.lookup_map_entry lookup_impl_of)
   144 
   145 lemma lookup_map [simp]:
   146   "lookup (map f t) k = Option.map (f k) (lookup t k)"
   147   by (simp add: map_def lookup_RBT RBT_Impl.lookup_map lookup_impl_of)
   148 
   149 lemma fold_fold:
   150   "fold f t = More_List.fold (prod_case f) (entries t)"
   151   by (simp add: fold_def fun_eq_iff RBT_Impl.fold_def entries_impl_of)
   152 
   153 lemma is_empty_empty [simp]:
   154   "is_empty t \<longleftrightarrow> t = empty"
   155   by (simp add: rbt_eq_iff is_empty_def impl_of_empty split: rbt.split)
   156 
   157 lemma RBT_lookup_empty [simp]: (*FIXME*)
   158   "RBT_Impl.lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
   159   by (cases t) (auto simp add: fun_eq_iff)
   160 
   161 lemma lookup_empty_empty [simp]:
   162   "lookup t = Map.empty \<longleftrightarrow> t = empty"
   163   by (cases t) (simp add: empty_def lookup_def RBT_inject RBT_inverse)
   164 
   165 lemma sorted_keys [iff]:
   166   "sorted (keys t)"
   167   by (simp add: keys_def RBT_Impl.keys_def sorted_entries)
   168 
   169 lemma distinct_keys [iff]:
   170   "distinct (keys t)"
   171   by (simp add: keys_def RBT_Impl.keys_def distinct_entries)
   172 
   173 
   174 end